Geometry · G.C.B.5

Arc length

You can find the distance all the way round a circle. How long is just a piece of the rim, cut off by an angle at the center?

You need

Arcs and central anglesGeometry

Where you are 15 in Circles

Geometry11

Circles15

▴ 5 above Arcs and central angles Area of a circle

Arc length

Sectors, segments and their areas The inscribed angle theorem ▾ Show all 15 (5 more)

When you reach for it

The Circumference formula, C = 2\pi r, gives the distance all the way round a circle. Often you need only part of the way round: the curved crust of a slice of pizza, the path of a clock hand’s tip over twenty minutes, the bend of a running track, the distance between two cities on the same line of longitude. Each of these is an arc, a piece of the circle between two of its points, and what you want is its arc length: the distance measured along the curve.

Reach for this move when you know how far round the circle the arc goes, as an angle at the center, and how big the circle is, as a radius or a diameter. It also runs backwards. Knowing an arc’s length and either one of the other two, you can find the third.

From arcs and central angles you already have one number attached to an arc: its measure, in degrees. This lesson adds a second, its length. The two describe different things, and the first job is to keep them apart.

The procedure

Arc measure and arc length are different quantities

The measure of an arc, written m\overset{\frown}{AB}, is in degrees. For a minor arc it equals the central angle \angle AOB that cuts it off, where O is the center, and for a major arc it is 360^\circ minus that angle. It says how far round the circle the arc goes, as a share of a full turn. It says nothing at all about how big the circle is.

The length of an arc, written s here, is a distance, in centimeters or meters or miles. It is how far you would walk going along the rim from one end of the arc to the other.

	Arc measure	Arc length
What it tells you	how far round, as a share of a full turn	the distance along the curve
Units	degrees	units of length: cm, m, km
The whole circle	360^\circ	the circumference, 2\pi r
Changes with the radius?	no	yes, in proportion to it

So “the arc is 72^\circ” and “the arc is 2\pi cm” are two different facts. Example 3 shows two arcs with the same measure and different lengths.

The length is a fraction of the circumference

Everything rests on one fact: in the same circle, equal central angles cut off arcs of equal length. The reason is a rotation about the center O. A rotation keeps every point at the same distance from O, so it carries the circle onto itself, and it is a rigid motion, so it changes no length. If \angle AOB and \angle COD are equal, named so that both open the same way round from their first ray, the rotation that turns ray OA onto ray OC also turns ray OB onto ray OD, so it lays the arc \overset{\frown}{AB} exactly on top of the arc \overset{\frown}{CD}, and the two arcs have the same length.

Now cut a circle with six radii, 60^\circ apart.

Figure 1Six equal central angles cut the rim into six arcs of equal length, so a 60^\circ arc is \displaystyle \frac{1}{6} of the circumference.

The six central angles are equal, so the six arcs are equal in length, and together they make the whole rim. Each one is therefore \displaystyle \frac{1}{6} of the circumference, and \displaystyle \frac{1}{6} is exactly \displaystyle \frac{60}{360}, the share of the full turn that its angle takes.

The same cut works for any angle. Three hundred and sixty radii one degree apart make 360 equal arcs, each \displaystyle \frac{1}{360} of the circumference, and an arc of n^\circ is made of n of them. For an angle that is not a whole number of degrees, cut into tenths or hundredths of a degree instead; the share stays \displaystyle \frac{n}{360}. So an arc is the same fraction of the circumference as its angle is of a full turn:

s = \frac{n}{360} \cdot 2\pi r

Here n is the arc’s measure in degrees and r is the radius. Two quick checks: n = 360 gives 2\pi r, the whole circumference, and n = 180 gives \pi r, half of it. Canceling the 2 against the 360 gives a shorter form of the same formula:

s = \frac{\pi n r}{180}

You do not have to use the formula as a formula. “A 120^\circ arc is a third of a turn, so it is a third of the circumference” is the same reasoning said aloud, and the examples below use whichever reads more clearly.

Same angle, bigger circle

Write the short form as \displaystyle s = \frac{\pi n}{180} \cdot r. For a fixed angle n, the factor \displaystyle \frac{\pi n}{180} is a fixed number, so arc length is proportional to the radius: double the radius and the arc doubles, triple it and the arc triples.

This is the similarity of circles at work. Every circle is a scaled copy of every other: a dilation centered at O with scale factor k carries a circle of radius r onto the circle of radius kr. A dilation keeps every angle and multiplies every length by k, so it carries an n^\circ arc to an n^\circ arc that is k times as long. The measure did not change; the length did.

The fixed factor \displaystyle \frac{\pi n}{180} is the arc length for each unit of radius. It depends only on the angle, which makes it a way of measuring the angle itself, and it is the angle’s measure in radians.

Working in either direction

The formula ties three quantities together: s, n and r. Know any two and the third is found by solving one equation. Given a diameter, halve it, or use C = \pi d directly. Keep \pi in the working and give the exact answer in terms of \pi; round only at the very end, since rounding \pi early carries the error through every later step.

Worked examples

Example 1

An arc from its central angle and radius

A circle has center O and radius 9 cm, and the central angle \angle AOB measures 80^\circ. Find the length of the minor arc \overset{\frown}{AB}, exactly and to two decimal places.

Solution

A minor arc’s measure is its central angle’s, so n = 80. The arc is \displaystyle \frac{80}{360} = \frac{2}{9} of a full turn, so it is \displaystyle \frac{2}{9} of the circumference. The circumference is 2\pi \cdot 9 = 18\pi cm, so

s = \frac{2}{9} \cdot 18\pi = 4\pi \text{ cm} \approx 12.57 \text{ cm}

Simplifying \displaystyle \frac{80}{360} first kept the arithmetic small. As a sense check, 80^\circ is a little less than a quarter turn, and a quarter of 18\pi \approx 56.55 is about 14.14, so an answer a little under 14 cm is right.

Example 2

A minor arc and its major arc

Points P and Q lie on a circle of diameter 18 m, and the minor arc \overset{\frown}{PQ} measures 100^\circ. A third point R lies on the major arc. Find the lengths of the minor arc \overset{\frown}{PQ} and of the major arc \overset{\frown}{PRQ}.

Solution

The diameter is given, so the circumference is C = \pi d = 18\pi m.

The minor arc is \displaystyle \frac{100}{360} = \frac{5}{18} of the turn:

s = \frac{5}{18} \cdot 18\pi = 5\pi \text{ m} \approx 15.71 \text{ m}

The major arc goes the long way round, so its measure is 360^\circ - 100^\circ = 260^\circ, which is \displaystyle \frac{260}{360} = \frac{13}{18} of the turn:

s = \frac{13}{18} \cdot 18\pi = 13\pi \text{ m} \approx 40.84 \text{ m}

The two arcs together are the whole circle, and indeed 5\pi + 13\pi = 18\pi. Notice that the angle \angle POQ is 100^\circ whichever arc you look at; the major arc’s length needs the major arc’s measure, 260^\circ, not the angle between the radii.

Example 3

Arc measure against arc length

Two circles share the center O and have radii 2 cm and 5 cm. Two rays from O make an angle of 72^\circ and cut off the arc \overset{\frown}{AB} on the small circle and the arc \overset{\frown}{CD} on the large one. Compare the measures of the two arcs, and compare their lengths.

Figure 2Both arcs measure 72^\circ, but the outer one is \displaystyle \frac{5}{2} times as long, because its radius is \displaystyle \frac{5}{2} times as long.

Solution

Measures. Both arcs are cut off by the same 72^\circ central angle, so m\overset{\frown}{AB} = m\overset{\frown}{CD} = 72^\circ. As shares of a full turn, the two arcs are the same.

Lengths. Both are \displaystyle \frac{72}{360} = \frac{1}{5} of their own circle’s circumference, but the circumferences differ: 4\pi cm and 10\pi cm.

\text{small: } \frac{1}{5} \cdot 4\pi = \frac{4\pi}{5} \approx 2.51 \text{ cm}

\text{large: } \frac{1}{5} \cdot 10\pi = 2\pi \approx 6.28 \text{ cm}

The large arc is \displaystyle 2\pi \div \frac{4\pi}{5} = \frac{5}{2} times as long, which is exactly the ratio of the radii, \displaystyle \frac{5}{2}. That is the similarity argument made concrete: the dilation centered at O with scale factor \displaystyle \frac{5}{2} carries the small circle onto the large one and each ray onto itself, so it carries A to C and B to D, keeps the 72^\circ, and stretches the arc by \displaystyle \frac{5}{2}.

So the measure is about how far round, and the length is about how far. Two arcs are congruent only when their lengths match too, which for equal measures means equal radii.

Example 4

Finding the angle from the length

An arc of a circle of radius 12 cm is 5\pi cm long. Find its measure.

Solution

The circumference is 2\pi \cdot 12 = 24\pi cm, so the arc is

\frac{5\pi}{24\pi} = \frac{5}{24}

of the whole circle. Its measure is the same share of 360^\circ:

n = \frac{5}{24} \cdot 360 = 75^\circ

The same thing done with the formula: \displaystyle 5\pi = \frac{n}{360} \cdot 24\pi. Dividing both sides by 24\pi gives \displaystyle \frac{n}{360} = \frac{5}{24}, and multiplying both sides by 360 gives n = 75. The \pi canceled, which is why a measure in degrees comes out with no \pi in it. Check: \displaystyle \frac{75}{360} \cdot 24\pi = \frac{5}{24} \cdot 24\pi = 5\pi.

Example 5

Finding the radius from an arc

An arc measuring 135^\circ is 9\pi inches long. Find the radius of the circle.

Solution

The arc is \displaystyle \frac{135}{360} = \frac{3}{8} of the turn, so 9\pi is \displaystyle \frac{3}{8} of the circumference. Write that as an equation and solve for r:

\begin{aligned} \frac{3}{8} \cdot 2\pi r &= 9\pi \\ \frac{3\pi r}{4} &= 9\pi \\ r &= 9\pi \cdot \frac{4}{3\pi} = 12 \end{aligned}

Multiplying both sides by \displaystyle \frac{4}{3\pi}, the reciprocal of the coefficient of r, is allowed because \displaystyle \frac{3\pi}{4} is not zero. The radius is 12 inches. Reasoning in words gives the same: if \displaystyle \frac{3}{8} of the circumference is 9\pi, then \displaystyle \frac{1}{8} of it is 3\pi, the whole is 24\pi, and 2\pi r = 24\pi gives r = 12.

Example 6

The tip of a clock hand

The minute hand of a clock is 15 cm long. How far does its tip travel between 3:00 and 3:20?

Figure 3From 3:00 to 3:20 the tip of the minute hand sweeps a 120^\circ arc of a circle of radius 15 cm.

Solution

The hand turns about the pivot at the center, and its tip stays 15 cm from that pivot, so the tip moves along a circle of radius 15 cm. The minute hand makes one full turn, 360^\circ, every 60 minutes, so in 20 minutes it turns

\frac{20}{60} \cdot 360^\circ = 120^\circ

That is \displaystyle \frac{1}{3} of a turn. The circumference of the tip’s circle is 2\pi \cdot 15 = 30\pi cm, so the tip travels

s = \frac{1}{3} \cdot 30\pi = 10\pi \text{ cm} \approx 31.42 \text{ cm}

Only the angle the hand turns through matters. The same twenty minutes starting at any other hour would give the same distance.

Example 7

Measuring the Earth

Two cities lie on the same meridian, the north to south line of longitude, 1000 km apart measured along the Earth’s surface. Their latitudes differ by 9^\circ. Treating the Earth as a sphere, estimate its circumference and its radius.

Solution

A meridian is half of a circle whose center is the center of the Earth, and a place’s latitude is an angle measured at that center, between the plane of the equator and the line out to the place. So the two latitudes are two rays from the Earth’s center, the angle between them is 9^\circ, and the 1000 km between the cities is the length of the arc they cut off.

This is the formula run backwards, with the circumference as the unknown. The arc is \displaystyle \frac{9}{360} = \frac{1}{40} of a full turn, so 1000 km is \displaystyle \frac{1}{40} of the circumference:

C = 40 \cdot 1000 = 40\,000 \text{ km}

Then C = 2\pi r gives the radius:

r = \frac{40\,000}{2\pi} = \frac{20\,000}{\pi} \approx 6366 \text{ km}

Both are close to the measured values, about 40\,000 km round the poles and a mean radius of about 6371 km. Nobody has to walk round the Earth to know its size: one measured arc and one measured angle are enough.

Boundaries

The angle must be at the center. The n in the formula is the arc’s measure, which equals a central angle. An angle with its vertex anywhere else does not give it directly. An inscribed angle, with its vertex on the circle, is half the arc it opens onto, by the inscribed angle theorem, so double it first (Exercise 19).

The measure must be in degrees. The share of a full turn is \displaystyle \frac{n}{360} only when n is in degrees. With an angle \theta in radians the share is \displaystyle \frac{\theta}{2\pi}, and the formula becomes s = r\theta, which is arc length in radians. A radian measure put into \displaystyle \frac{n}{360} gives a wrong answer.

A major arc needs its own measure. The angle between the two radii is the same for both arcs. The major arc’s measure is 360^\circ minus the minor arc’s, as in Example 2.

An arc’s measure lies between 0^\circ and 360^\circ. A turn of more than 360^\circ is not an arc any more, but the distance a point on the rim travels while the circle turns that far is still \displaystyle \frac{n}{360} of the circumference, as with the rolling wheel in Exercise 17.

The arc is not the chord. The straight segment AB is shorter than the arc \overset{\frown}{AB}, since a straight segment is the shortest path between two points. Finding a chord’s length is a triangle problem, not this one.

Circles only. The argument needed every point of the curve to be the same distance from one center. An ellipse or a parabola has no such center, and the length along a general curve takes calculus.

The answer has the radius’s units. The degrees cancel in \displaystyle \frac{n}{360}, so a radius in centimeters gives an arc in centimeters.

Where this goes next

The same share of the turn, applied to the area \pi r^2 instead of the circumference, gives the area of a sector: the slice of pizza whose crust this lesson measured.

The fact that, for a fixed angle, the arc length divided by the radius is always the same number is the whole idea of radian measure. Measure angles that way and the formula loses its \displaystyle \frac{1}{360} and its 2\pi and becomes simply s = r\theta.

Example 6 measured a distance traveled in a given time. Divide the distance by the time and you have the linear speed of the tip; divide the angle turned by the time instead and you have angular speed. This lesson’s formula is what links the two, through the radius. And for curves that are not circles, the calculus of arc length adds up tiny straight pieces along the curve, which for a circle gives back exactly this formula.

Exercises

Find the arc length

Give each answer exactly, in terms of \pi, and to two decimal places.

Radius 10 cm, central angle 36^\circ.
Radius 4 in, central angle 225^\circ.
Diameter 14 m, central angle 60^\circ.
Radius 5 ft, central angle 150^\circ.
Radius 2.5 cm, central angle 18^\circ.

Minor and major arcs

In a circle of radius 6 cm, the minor arc \overset{\frown}{AB} measures 110^\circ. Find the measure of the major arc, and the length of each arc.
Points A, B and C lie in that order round a circle of radius 9 cm, with m\overset{\frown}{AB} = 70^\circ and m\overset{\frown}{BC} = 130^\circ, neither arc containing the third point. Find the measure and the length of the arc \overset{\frown}{CA} that does not contain B.

Arc measure and arc length

Arc \overset{\frown}{PQ} lies on a circle of radius 3 cm and arc \overset{\frown}{RS} on a circle of radius 9 cm, and both measure 40^\circ. Find the length of each. How many times as long as \overset{\frown}{PQ} is \overset{\frown}{RS}, and why?
Which is longer: a 90^\circ arc of a circle of radius 4 cm, or a 30^\circ arc of a circle of radius 10 cm?
Which is longer: a 120^\circ arc of a circle of radius 3 cm, or a 45^\circ arc of a circle of radius 10 cm?

Working backwards

An arc of a circle of radius 8 cm is 6\pi cm long. Find its measure.
A 40^\circ arc is 10\pi m long. Find the radius of its circle.
A 150^\circ arc is 20 cm long. Find the radius of its circle, exactly and to two decimal places.
On one circle, the minor arc \overset{\frown}{AB} is 8\pi cm long and the major arc \overset{\frown}{ACB} is 28\pi cm long. Find the radius and m\overset{\frown}{AB}.
A 45^\circ arc of a circle is 7 cm long. Find the circumference of the circle.

Problems

The hour hand of a clock is 6 cm long. How far does its tip travel from 12:00 to 5:00?
A bicycle wheel of diameter 70 cm rolls along flat ground without slipping while it turns through 900^\circ. How far does the bicycle move?
A slice is cut from a round pizza of radius 15 cm, with a central angle of 48^\circ. Find the perimeter of the slice: its two straight edges and its crust.
Points A, B and C lie on a circle of radius 9 cm, and the inscribed angle \angle ACB measures 35^\circ. Find the length of the arc \overset{\frown}{AB} that does not contain C.
A pendulum 80 cm long swings from one side to the other through an angle of 25^\circ. How long is the path of the bob in one swing?

Reasoning

Explain the difference between an arc’s measure and its length. Give two arcs with equal measures and different lengths, and two arcs with equal lengths and different measures.
Explain why, in one circle, two equal central angles cut off arcs of equal length, and use this to explain why an arc of n^\circ is \displaystyle \frac{n}{360} of the circumference.
Show that \displaystyle \frac{n}{360} \cdot 2\pi r = \frac{\pi n r}{180}. Use it to explain why doubling the radius with the angle fixed doubles the arc length, and why doubling the angle with the radius fixed doubles it too.
An arc is exactly as long as the radius of its circle. Find its measure in degrees, exactly and to two decimal places, and explain why the answer is the same for every circle.

Answers

2\pi \approx 6.28 cm. The arc is \displaystyle \frac{36}{360} = \frac{1}{10} of the circumference 20\pi.
5\pi \approx 15.71 in. The arc is \displaystyle \frac{225}{360} = \frac{5}{8} of the circumference 8\pi.
\displaystyle \frac{7\pi}{3} \approx 7.33 m. The circumference is 14\pi, and \displaystyle \frac{60}{360} = \frac{1}{6} of it is \displaystyle \frac{14\pi}{6}.
\displaystyle \frac{25\pi}{6} \approx 13.09 ft. The arc is \displaystyle \frac{150}{360} = \frac{5}{12} of the circumference 10\pi.
\displaystyle \frac{\pi}{4} \approx 0.79 cm. The arc is \displaystyle \frac{18}{360} = \frac{1}{20} of the circumference 5\pi.
The major arc measures 250^\circ. Minor arc: \displaystyle \frac{11\pi}{3} \approx 11.52 cm; major arc: \displaystyle \frac{25\pi}{3} \approx 26.18 cm. The circumference is 12\pi; \displaystyle \frac{110}{360} = \frac{11}{36} and \displaystyle \frac{250}{360} = \frac{25}{36} of it. The two lengths add to 12\pi.
m\overset{\frown}{CA} = 160^\circ, length 8\pi \approx 25.13 cm. The three arcs make the whole circle, so the measure is 360^\circ - 70^\circ - 130^\circ, and \displaystyle \frac{160}{360} = \frac{4}{9} of 18\pi is 8\pi.
\overset{\frown}{PQ} is \displaystyle \frac{2\pi}{3} \approx 2.09 cm and \overset{\frown}{RS} is 2\pi \approx 6.28 cm, so \overset{\frown}{RS} is 3 times as long. Both are \displaystyle \frac{1}{9} of their circle’s circumference, 6\pi and 18\pi; with the angle fixed, arc length is proportional to the radius, and 9 = 3 \cdot 3.
The 90^\circ arc. It is \displaystyle \frac{1}{4} of 8\pi, so 2\pi \approx 6.28 cm; the 30^\circ arc is \displaystyle \frac{1}{12} of 20\pi, so \displaystyle \frac{5\pi}{3} \approx 5.24 cm.
The 45^\circ arc, even though its measure is smaller. It is \displaystyle \frac{1}{8} of 20\pi, so \displaystyle \frac{5\pi}{2} \approx 7.85 cm; the 120^\circ arc is \displaystyle \frac{1}{3} of 6\pi, so 2\pi \approx 6.28 cm.
135^\circ. The circumference is 16\pi, so the arc is \displaystyle \frac{6\pi}{16\pi} = \frac{3}{8} of the circle, and \displaystyle \frac{3}{8} \cdot 360 = 135.
45 m. The arc is \displaystyle \frac{40}{360} = \frac{1}{9} of the circumference, so the circumference is 90\pi, and 2\pi r = 90\pi.
\displaystyle r = \frac{24}{\pi} \approx 7.64 cm. \displaystyle \frac{150}{360} \cdot 2\pi r = \frac{5\pi r}{6} = 20, so \displaystyle r = 20 \cdot \frac{6}{5\pi}.
Radius 18 cm, m\overset{\frown}{AB} = 80^\circ. The two arcs make the circle, so the circumference is 8\pi + 28\pi = 36\pi and r = 18. The minor arc is \displaystyle \frac{8\pi}{36\pi} = \frac{2}{9} of the circle, and \displaystyle \frac{2}{9} \cdot 360 = 80.
56 cm. A 45^\circ arc is \displaystyle \frac{45}{360} = \frac{1}{8} of the circle, so the circumference is 8 \cdot 7.
5\pi \approx 15.71 cm. The hour hand turns 360^\circ in 12 hours, so in 5 hours it turns 150^\circ, which is \displaystyle \frac{5}{12} of the circumference 12\pi.
175\pi \approx 549.78 cm, about 5.5 m. Each full turn rolls out one circumference, 70\pi cm, since every point of the rim touches the ground once; 900^\circ is \displaystyle \frac{900}{360} = \frac{5}{2} turns.
30 + 4\pi \approx 42.57 cm. The straight edges are two radii, 30 cm together, and the crust is \displaystyle \frac{48}{360} = \frac{2}{15} of 30\pi, which is 4\pi.
\displaystyle \frac{7\pi}{2} \approx 11.00 cm. By the inscribed angle theorem the arc measures twice the inscribed angle, 70^\circ, and \displaystyle \frac{70}{360} \cdot 18\pi = \frac{7\pi}{2}.
\displaystyle \frac{100\pi}{9} \approx 34.91 cm. The bob moves on a circle of radius 80 cm about the pivot, and \displaystyle \frac{25}{360} \cdot 160\pi = \frac{100\pi}{9}.
The measure is the share of a full turn, in degrees, and is the same however big the circle is; the length is a distance along the curve and grows with the radius. A 60^\circ arc of a circle of radius 1 (length \displaystyle \frac{\pi}{3}) and a 60^\circ arc of radius 3 (length \pi) have equal measures and different lengths. A 90^\circ arc of radius 4 and a 180^\circ arc of radius 2 both have length 2\pi, with different measures.
A rotation about the center keeps every point at the same distance from the center, so it maps the circle onto itself, and it preserves length. The rotation taking one radius of the first angle onto a radius of the second takes the whole first angle onto the second, so it lays one arc on the other, and they have equal length. Then 360 equal central angles of 1^\circ cut the circle into 360 equal arcs, each \displaystyle \frac{1}{360} of the circumference, and an n^\circ arc is n of them, so it is \displaystyle \frac{n}{360} of the circumference.
\displaystyle \frac{n}{360} \cdot 2\pi r = \frac{2\pi n r}{360} = \frac{\pi n r}{180}, dividing top and bottom by 2. The arc length is a constant times n r: with n fixed, replacing r by 2r doubles the product, and with r fixed, replacing n by 2n doubles it too. Doubling the angle keeps working only while 2n is still at most 360.
\displaystyle \frac{180}{\pi} \approx 57.30 degrees. Setting s = r in \displaystyle s = \frac{\pi n r}{180} gives \displaystyle r = \frac{\pi n r}{180}, and dividing both sides by r (not zero) gives \displaystyle n = \frac{180}{\pi}. The radius canceled, so every circle gives the same angle; this angle is one radian.